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So for f E £,1/2,1 we have 'P E Coo(lR) (in fact 'P E B~l, which is slightly better locally but not so explicit), while for f E £,s,p with s > 1/2 we have 'P E As- 1/ 2 ,p, and this is optimal on the scale {A,8,q}. In terms of the Lipschitz-Zygmund scale {N~}, the best one can have for the boundary value function 'P is: 'P E AS- 1/ 2 if f E £,s,p and s > 1/2. For example, if s = 3/2, then 'P E A I and (in general) 'P is not a locally Lipschitz continuous function unless p = 1, in which case 'P is of class BC I (cf.

Belongs to a compact subset of R. \ {O}. We have denoted by II . )1/2,1. 10) remains true if II . II(s1) is replaced by II . 11(1/2,1) and 82 > 1/2. Summarizing, we have proved that for each real 8 > 1/2 and >. #- 0 the operators R(>. 9) is locally of class As-1/2. g. [Q, iPJ = -1. 36 ANNE BOUTET DE MONVEL AND VLADIMIR GEORGESCU 4. 1. This section is entirely devoted to the proof of the theorem stated in the Introduction. Yt'). We choose a real even function () E Co (lR) , so that ()+(x) : : : : ()(-x) = ()(x).